The answer seems to be that microbes (bacteria, archaea etc) float around in the ocean currents in their billions. When a new vent forms, or an existing vent is cleared of its existing residents by an eruption, new microbes that float by latch on and grow in the warmth into a thin biofilm that covers the rocks.
The larvae of bigger fauna (like mussels, limpets, tube worms etc) also float up in the warm water plume from their home vent and then get dispersed into the ocean currents in their billions, often floating ”downstream” from other nearby vents, but sometimes from hundreds of miles away. Those that are lucky enough to settle on the biofilm at the new vent start to colonise and transform from their larval stage into adults which can only survive in that adult form in the warmer environment.
See [https://www.sciencedirect.com/science/article/pii/S0967063724000840](https://www.sciencedirect.com/science/article/pii/S0967063724000840) and [https://www.pnas.org/doi/full/10.1073/pnas.0913187107](https://www.pnas.org/doi/full/10.1073/pnas.0913187107)
/not a biologist
ETA: There is an article here that says that larvae of the giant tubeworm can survive for up to 38 days in ocean currents away from the warmth, presumably long enough to float their way to a new vent.
[https://www.whoi.edu/press-room/news-release/colonizing-the-deep-sea-whoi-scientist-helps-find-answers-to-hydrothermal-vent-puzzle/](https://www.whoi.edu/press-room/news-release/colonizing-the-deep-sea-whoi-scientist-helps-find-answers-to-hydrothermal-vent-puzzle/)
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